Prove that `(1)/(sqrt3)` is irrational.

Answer

To prove that \( \frac{1}{\sqrt{3}} \) is irrational, we can follow these steps:

### Step 1: Assume the opposite
Assume that \( \frac{1}{\sqrt{3}} \) is a rational number. By definition, a rational number can be expressed as the ratio of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

### Step 2: Rewrite the assumption
If \( \frac{1}{\sqrt{3}} \) is rational, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{p}{q} \]
This implies:
\[ 1 = \frac{p}{q} \cdot \sqrt{3} \]
or
\[ \sqrt{3} = \frac{q}{p} \]

### Step 3: Analyze the implications
From the equation \( \sqrt{3} = \frac{q}{p} \), we see that \( \sqrt{3} \) is expressed as a ratio of two integers \( q \) and \( p \). This means that \( \sqrt{3} \) is rational.

### Step 4: Recall the nature of \( \sqrt{3} \)
However, it is a well-known fact that \( \sqrt{3} \) is an irrational number. This leads to a contradiction because we cannot have both \( \sqrt{3} \) being rational and irrational at the same time.

### Step 5: Conclude the proof
Since our assumption that \( \frac{1}{\sqrt{3}} \) is rational leads to a contradiction, we conclude that our initial assumption must be false. Therefore, \( \frac{1}{\sqrt{3}} \) is irrational.